The position Vectors of two identical particles with respect to the origin in the three-dimensional coordinator system are `r_1and r_2` The position of the centre of mass of the system is given by

Options

A. (a)`r_1 +r_2`
B. (b)`2 (r_1+r_2)`
C. (c)`r_1 -r_2`
D. (d)`(r_1+r_2)/2`

Answer

To find the position of the center of mass of two identical particles with position vectors \( \mathbf{r_1} \) and \( \mathbf{r_2} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The position vectors of the two identical particles are given as \( \mathbf{r_1} \) and \( \mathbf{r_2} \).
- Since the particles are identical, we denote their masses as \( m_1 = m \) and \( m_2 = m \).

2. **Use the Formula for Center of Mass:**
- The formula for the position vector of the center of mass \( \mathbf{R} \) of a system of particles is given by:
\[ \mathbf{R} = \frac{m_1 \mathbf{r_1} + m_2 \mathbf{r_2}}{m_1 + m_2} \]

3. **Substitute the Masses:**
- Since both particles have the same mass \( m \), we can substitute \( m_1 \) and \( m_2 \) in the formula:
\[ \mathbf{R} = \frac{m \mathbf{r_1} + m \mathbf{r_2}}{m + m} \]

4. **Simplify the Expression:**
- Factor out the mass \( m \) from the numerator:
\[ \mathbf{R} = \frac{m (\mathbf{r_1} + \mathbf{r_2})}{2m} \]
- The mass \( m \) in the numerator and denominator cancels out:
\[ \mathbf{R} = \frac{\mathbf{r_1} + \mathbf{r_2}}{2} \]

5. **Conclusion:**
- The position vector of the center of mass of the two identical particles is:
\[ \mathbf{R} = \frac{\mathbf{r_1} + \mathbf{r_2}}{2} \]